Numerical investigation of the sharp-interface limit of the Navier-Stokes-Cahn-Hilliard equations

In this article, we study the behaviour of the Abels-Garcke-Grün Navier-Stokes-Cahn- Hilliard diffuse-interface model for binary-fluid flows, as the diffuse-interface thickness passes to zero. For the diffuse-interface model to approach a classical sharp-interface model in the limit, the so-called mobility parameter in the diffuse-interface model must scale appropriately with the interface-thickness parameter. In the literature various scaling relations in the range to have been proposed, but the optimal order to pass to the limit has not been explored previously. Our primary objective is to elucidate this optimal order of the - scaling relation in terms of the rate of convergence of the diffuse-interface solution to the sharp-interface solution. Additionally, we examine how the convergence rate is affected by a sub-optimal parameter scaling. We centre our investigation around the case of an oscillating droplet. To provide reference limit solutions, we derive new analytical expressions for small-amplitude oscillations of a viscous droplet in a viscous ambient fluid in two dimensions. For two distinct modes of oscillation, we probe the sharp-interface limit of the Navier-Stokes-Cahn-Hilliard equations by means of an adaptive finite-element method. The adaptive-refinement procedure enables us to consider diffuse-interface thicknesses that are significantly smaller than other relevant length scales in the droplet-oscillation problem, allowing an exploration of the asymptotic regime.